contents · determination of the mcmahon bar ff kaniuk geffo.uk august 2015 contents 1...
TRANSCRIPT
DETERMINATION OF THE McMAHON BAR
Geoff Kaniuk [email protected]
August 2015
Contents
1 INTRODUCTION 5
2 FLAWED TOURNAMENTS 7
3 RANDOM PAIRING 10
3.1 Player Opponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Correlation between crossing points for ∆S and ∆G . . . . . . . 12
3.4 The equilibrium grade . . . . . . . . . . . . . . . . . . . . . . . . 14
4 SWISS PAIRING 15
4.1 Pairing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Crossing point correlation . . . . . . . . . . . . . . . . . . . . . . 16
5 McMAHON PAIRING 17
6 TOP GROUP PEER GAMES 19
7 OPPONENT GRADES 20
8 GROUP SCORES 22
9 SCORE DISTRIBUTIONS AND THE BAR 24
9.1 Player score and Group score . . . . . . . . . . . . . . . . . . . . 24
9.2 Winning chances above the bar . . . . . . . . . . . . . . . . . . . 25
9.3 Bar separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
10 BAR DETERMINATION METHOD 28
10.1 Statement of the method . . . . . . . . . . . . . . . . . . . . . . . 28
10.2 Scope of the Monte Carlo trial . . . . . . . . . . . . . . . . . . . 29
10.3 Solution failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.4 Variation with number of rounds . . . . . . . . . . . . . . . . . . 30
10.5 Tournament entry . . . . . . . . . . . . . . . . . . . . . . . . . . 30
10.6 Pairing sample rate . . . . . . . . . . . . . . . . . . . . . . . . . . 31
10.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1
11 MONTE-CARLO TRIAL RESULTS 32
11.1 Solution failures . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
11.2 Solution correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
11.3 Bar Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
11.4 Bar Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.5 Bar Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
11.6 Uniqueness and ranking . . . . . . . . . . . . . . . . . . . . . . . 37
12 GUIDELINES 38
12.1 Classical guidelines and maxims . . . . . . . . . . . . . . . . . . . 38
12.2 Additional guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 39
13 SUMMARY AND CONCLUSION 40
13.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
13.2 Key quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
13.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A MODEL FOR BAR SEPARATION 42
B PROBABILITY OF WIN 42
C GLOSSARY 43
D NOTATION 45
E ALGORITHM INDEX 45
List of Tables
1 McMahon bar settings . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Probability for flawed tournaments . . . . . . . . . . . . . . . . . 10
3 Entry for Tr-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Entry for Ts-gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Limits for the tournament entry . . . . . . . . . . . . . . . . . . . 30
6 Tm-gap entry 5 rounds 42 players . . . . . . . . . . . . . . . . . . 31
7 Parameter accuracy dependent on number of pairings . . . . . . 32
8 Outlier features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9 Tmcmahon entry 7 rounds 50 players . . . . . . . . . . . . . . . . . 35
10 Coefficients for winning probability . . . . . . . . . . . . . . . . . 43
2
List of Figures
1 Four players - pairing for round one . . . . . . . . . . . . . . . . 7
2 Four players - start of round two . . . . . . . . . . . . . . . . . . 8
3 Possible graphs for 6 players at start of round 3 . . . . . . . . . . 9
4 Opponent grade difference for Tideal . . . . . . . . . . . . . . . . 10
5 Mean group score for tournament Tideal . . . . . . . . . . . . . . 12
6 Scatter plot for ZS vs ZG in random-pairing . . . . . . . . . . . 13
7 Solution process for Tr-gap . . . . . . . . . . . . . . . . . . . . . 14
8 Swiss forms for ∆G and ∆S in Tideal . . . . . . . . . . . . . . . 15
9 Scatter diagram for ZS vs ZG in swiss-pairing . . . . . . . . . . 16
10 Solution process for Ts-gaps . . . . . . . . . . . . . . . . . . . . . 17
11 McMahon pairing quality in Tideal . . . . . . . . . . . . . . . . . 18
12 Peer Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
13 McMahon opponent grade difference . . . . . . . . . . . . . . . . 21
14 ∆G on the bar-layer . . . . . . . . . . . . . . . . . . . . . . . . . 22
15 Mean group-score for McMahon . . . . . . . . . . . . . . . . . . 23
16 Group-score and differences on the bar-layer . . . . . . . . . . . 23
17 Shodan player-score and group-score histograms . . . . . . . . . 25
18 Tail distributions T (g, 0, s) for bar at shodan . . . . . . . . . . . 26
19 The mid-score tail distributions Tmid(g, b) . . . . . . . . . . . . . 27
20 Mid score tail distribution Tmid(g, b) . . . . . . . . . . . . . . . . 27
21 Sampling at 10 and 100 pairings for Tm-gap . . . . . . . . . . . . 31
22 Scatter plot for ZS vs ZG in mcmahon-pairing . . . . . . . . . . 33
23 The Toutlier models . . . . . . . . . . . . . . . . . . . . . . . . . 33
24 Bar depth statistics . . . . . . . . . . . . . . . . . . . . . . . . . 34
25 Solution process and ramp models for Tmcmahon . . . . . . . . . 35
26 Ramp statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
27 Cumulative distribution for the bar population . . . . . . . . . . 36
28 Bar table statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 37
29 Unique winner probability and rank deviation for Tm-gap . . . . 38
3
References
[1] http://senseis.xmp.net/?McMahonPairing/BarTheory
[2] http://www.kaniuk.co.uk/articles/pwin/prob-win.pdf
[3] http://www.kaniuk.co.uk/articles/tourstats/tourstats.pdf
[4] https://en.wikipedia.org/wiki/Round-robin tournament
[5] Harold.N Gabow, Implementation of Algorithms for maximum matching onnonbipartite graphs(chapters 1 and 4),PhD Thesis,Stanford University,1974
[6] https://en.wikipedia.org/wiki/Table of simple cubic graphs#6 nodes
[7] http://www.kaniuk.co.uk/articles/pairing/mcmahon-weights-revised.pdf
[8] http://vannier.info/jeux/download/GothaHelp en.pdf
[9] Paul L Meyer, Introductory Probability and Statistical Applications,Addison-Wesley, 1972
[10] https://en.wikipedia.org/wiki/Cumulative distribution function
[11] https://en.wikipedia.org/wiki/Ramp function
[12] http://www.kaniuk.co.uk/source/
[13] http://www.kaniuk.co.uk/articles/pairing/mcmahon-bar-data.tar.bz2
[14] http://www.britgo.org/organisers/handbook/tournament4.html
4
1 INTRODUCTION
In a McMahon tournament [1], all players with grades above a certain gradecalled the bar, are given the same initial mcmahon-score. The bar is designedto satisfy three criteria:
1. There should be a unique winner.
2. Every player who has a reasonable chance of winning should be above thebar.
3. Top players should not run out of evenly matched opponents before theend of the tournament.
Since the McMahon system pairs players on the same mcmahon-score, theunique winner requirement can lead to an excessively large pool of players abovethe bar. For example, in an 8 round tournament, the pool would be 256 players- much larger than the usual total entry! For a large pool we generally find thatthere is no unique winner based on pure mcmahon-score, and indeed the topplayers may never meet. Tie-breaks are then needed to determine the winner.
In order to meet the second requirement, the bar is raised to include only thosewho have a reasonable chance of winning. However, if the pool is too small, thetop players run out of opponents well before the end of the tournament, andwill be playing low quality games against weaker players.
Most Go organisations have tables for setting the bar, defining the populationrange based on the number of rounds. Since it is not easy to pin down a valuefor ’a reasonable chance of winning’, it comes as no surprise that there is a greatdeal of variety in the tables.
rounds bga macmahon aga egf
3 8 7 8 -4 10 9 12 -5 12 13 18 166 15 17 24 247 18 21 32 308 22 25 50 409 26 29 - 5010 30 33 - 60
Table 1: McMahon bar settings
The McMahon Bar Theory article in Sensei’s Library [1] provides references tovarious tables. Upper limits for the bar population are summarised in Table 1.For any number of rounds, a larger bar population will result in a wider rangeof player grades above the bar. The weakest players in this range will normallyhave little chance of winning the tournament.
5
One of the first issues to resolve is how do we quantify the requirement thatplayers above the bar have a ’reasonable chance’ of winning the tournament.In the ideal case, the winner of the tournament wins all games played. Thissuggests that we start by examining the score distribution of players in thegroups above the bar. Highest graded players inevitably meet lower gradedplayers (also above the bar), and so have a greater probability of winning alltheir games.
The purpose of this study is to extract features of McMahon tournamentsdependent on details of the player entry, which may be useful for setting thebar. The probability distribution of scores in each group is one such feature andis dependent on:
� The probability of win between players of differing grade.
� The population of players at each grade.
� The pairing algorithm.
The win probability between players is one of the most significant pieces of datacollected in the European Go Database (E.G.D), and a suitable model for therepresentation of the win probability is presented in [2]. It is accurate in therange 12 kyu to 8 dan, and this model is used to simulate the results of gamesfor all the statistics presented in this document. We assume at the outset thatplayer strength is completely specified by player grade.
The distribution of player grades entering tournaments in Europe has beenstudied in tournament statistics [3] harvested from E.G.D. Models enabling usto generate plausible distributions have been derived, but in the first instanceit will be useful to consider the case of purely random distributions of playersin the range 8 kyu to 5 dan.
As far as the pairing algorithm is concerned, there are a variety of strategiesin use. One of the simplest is the Swiss system, and this has a bearing onthe behaviour of bar players in a McMahon tournament. There are howeverdifferences, as the Swiss tournament is closed, but in a McMahon tournamentplayers above the bar continually meet winners from grades below the bar.
Perhaps the simplest possible pairing method, and one that is not normallyconsidered, is pure random-pairing. There are of course good reasons whyrandom-pairing is not used, but it should be possible to gain some insight intothe behaviour of player scores in any group.
By pairing randomly there is the danger that the tournament may not becompleted, at least without some players having repeat games or byes. Thisproblem however is not restricted to random-pairing - the danger is there in anysystem especially if the ratio of players to number of rounds is below 2. In mostcases it is definitely taboo for players to have repeat games, and there may notbe enough time to arrange for the extra rounds that would be needed to pairplayers that received a bye.
6
A second issue with unstructured pairing is that the winner may well be decidedby lottery. This again occurs in other pairing systems where tie-breaks need tobe invoked to decide a winner. On both issues the question to be asked is: howbad is the effect? If random-pairing really is the worst kind of method, then atleast we will have developed a sound basis for comparison.
The plan of this study is to start by examining the case of random-pairing in thecontext of random distributions of player grades. The purpose of this is simplyto provide a framework for understanding how the player score distributions areaffected by variations in entry grades. We next examine swiss-pairing and thenmcmahon-pairing for small tournaments to gauge the effect of the feed fromplayers below the bar.
In all the tournaments discussed throughout this study, it is assumed that thereis an even number of players - so in principle there are no byes! Unless explicitlystated otherwise, random means that samples are drawn from a discrete uniformdistribution over a finite range.
Technical terms are italicised when first introduced, and summarised in theGlossary - see Appendix C. It is convenient when referring to player grades toexpress these in zero-shodan units. So a shodan is represented by 0G and a 1kyu player by −1G. A group refers to the set of all players of the same grade.The term group g denotes the set consisting of all players with grade g.
2 FLAWED TOURNAMENTS
Like all-play-all , random-pairing matches players blindly without regard to anygame information, other than excluding repeat games. Figure 1 represents thepairing for round 1 in a 3 round all-play-all tournament with 4 players:
r rr
r
����������
TT
TT
TT
TT
TT
""""""
bb
bb
bb1
23
4
Figure 1: Four players - pairing for round one
The vertices in Figure 1 (numbered 1 · · · 4) identify the players. The edgesjoining the vertices indicate the allowed pairs. The heavy edges represent thepairing, so 1 plays 4 and 2 plays 3.
For the second round, we remove the edges for the pairing in round 1 to ensurethat there are no repeat games. Then, following the standard Round-Robin
7
r rr
r
TT
TT
TT
TT
TT
""""""
1
42
3
Figure 2: Four players - start of round two
algorithm [4], we rotate the diagram clockwise to produce Figure 2. The graphis a cycle of 4 edges, so there are two possible pairings for round 2: 1-3 and 2-4(the standard round-robin pairing) or 1-2 and 3-4. Either way, the pairing isperfect (there are no byes) and we can complete all rounds of the tournament,even if we had chosen the pairing at random in each round.
All pairing algorithms operate along the following lines: we present the algorithmwith a list of edges defining the allowed pairings between players. It then choosesa matching (a set of non-intersecting edges). This ensures that no player hastwo opponents in the same round.
For more rounds, it may happen some players would be given a bye. Let uscall this a flawed tournament. Monte Carlo simulation is used to examine howfrequently random-pairing might result in a flawed tournament :
Algorithm 1. simulate-flawed-tournaments
TS1. Choose an odd number of rounds r in the range 3 to 15. Then set thenumber N of players to be r + 1.
TS2. Set up a list of edges allowing each player to play any other player.
TS3. Pair all the players at random.
TS4. If there are any byes, the tournament is flawed, and we continue at TS1.
TS5. Remove paired players from the list of edges. If the list is not emptycontinue at TS3.
TS6. When the list is empty, the tournament completes without flaw. Continueat TS1.
A maximum cardinality matching algorithm cmatch [5] is used in the abovesimulation in step TS3. Such algorithms try to find a matching with the largestnumber of edges. Pairing at random is achieved by shuffling the list of edgespresented to cmatch at each round.
Step TS1 is repeated 10 million times to ensure that the measured value for theprobability of finding a flawed tournament is accurate to better than 0.1%. In
8
most cases where the pairing was not perfect, the flaw occurs at the penultimateround (indicated by the column -1R) as shown in Table 2. There are a very fewtournaments where the flaw does occur at 3 rounds prior to the last.
For six players it is not difficult to see why there is the possibility of a flawedtournament at round four. Let us go back to the beginning of round three.
At this point each player has just three possible opponents. There are exactly[6] two possible graphs1 representing this situation as shown in Figure 3.
SS
SS
SS
��
��
��
������
SSSSSSs s
s s
s s
1 2
3 4
5 6
!!!!!!!!
��
��
��
���
aaaaaaaa
SSS
SSSSSS
s s
s s
s
s
1 2
3 4
5
6
Figure 3: Possible graphs for 6 players at start of round 3
For the left hand diagram, one possible matching is 1-3, 2-4, 5-6, and whenthese edges are removed at the start of round 4, we are left with the cycle1-5-3-4-6-2-1 of six edges. This means that there are two perfect pairings forrounds 4 and 5.
The other possible matching is 1-2, 3-4, 5-6, and now we see that this would bea bad choice. For at the start of round 4 we are left with two separated triangles1-3-5 and 2-4-6, and only two players in each triangle can be paired .
In the right hand diagram choose the matching 1-3, 2-4, 5-6 and remove theseedges from the graph. We are left with the cycle 1-2-5-3-4-6-1, which meansthat we can continue to pair the tournament for rounds 4 and 5. In fact, theright hand graph always has perfect matchings for the next three rounds, nomatter which is chosen for round three.
The simulation covered a large number of matchings, but not every conceivablepairing method is encountered. I am indebted to Fred Holroyd and Tim Huntfor the example of a tournament with 10 players paired by the split-and-cyclemethod sometimes used in team tournaments.
Players are split into two equal teams A and B; one team remains seated andthe other cycles through board positions at each round. This pairing methodcan continue happily for 5 rounds, but at round 6 every player in team A hasplayed every player in team B.
Team A has 5 players and only 4 of them can now be paired. The same situationholds for team B, so we are left with two byes and the tournament is flawed.
1I thank Richard Parker for pointing me in the direction of cubic graphs.
9
players -3R -2R -1R 0R perfect
4 - - - - 1.0006 0 0 0.150 0 0.8508 0 0 0.156 0 0.84410 3.5× 10−6 0 0.213 0 0.78712 0 0 0.247 0 0.75314 0 0 0.277 0 0.72316 7.0× 10−7 0 0.302 0 0.698
Table 2: Probability for flawed tournaments
3 RANDOM PAIRING
3.1 Player Opponents
The pairing algorithm decides each player’s opponent, and in a random-pairingtournament the opponent can have almost any grade. There is a small bias inthe choice of opponents, as players are not allowed repeated pairings, but inaddition, players with the lowest grade gmin will play others of the same gradeor stronger.
Let G(g) be the average grade of the opponents of group g. It is highlylikely that G(gmin) > gmin in a randomly paired tournament, as the weakestplayers will sometimes meet stronger players. On the other hand, the strongestplayers will meet opponents of the same grade gmax or weaker, so we can expectG(gmax) < gmax.
∆G
grade
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 4: Opponent grade difference for Tideal
We define the opponent-grade-difference as:
∆G(g) = G(g)− g (1)
10
Here X is the mean taken over simulations of pairings repeated many timeson the same entry. From the discussion above, ∆G(g) varies from positive atg=gmin to negative at g=gmax.
Consider the highly idealised 6 round tournament Tideal, with 4 players in eachgrade varying from −8G to 4G. We simulate this tournament in the same wayas described in Algorithm simulate-flawed-tournaments, from steps TS2 toTS6. Since there are many more players than number of rounds, we do not seeany flawed tournaments, even with 1000 simulations of the same entry.
The behaviour of ∆G(g) for all trial pairings is shown in Figure 4. The linearityis expected2 given the uniform entry. This clearly has a single crossing point atgrade Gc=−2G; these players enjoy a special status in that they play others oneither side of their own grade to get a balanced set of opponents.
3.2 Score
Once the pairing for a round has been decided, it is up to the players to scoreas best they can. The outcome is statistically determined by the probability ofwin between players of given, possibly different, grades gi, gj . A game result issimulated by:
Algorithm 2. simulate-result
RS 1. Calculate the probability Pij = Pwin(gi, gj) that player i beats j.
RS 2. Choose a real number f in the range [0, 1].
RS 3. Assuming gi < gj , the result of the game is a win for i if f < Pij .
Appendix B summarises the detail of the calculation for Pwin(gi, gj). At theend of each simulated tournament we obtain the total score for each player, andwe define the group-score for group g to be:
s(g) =1
ng
ng∑i=1
wi
Here ng is the number of players in the group, and player i has wi wins. At theend of a randomly paired tournament with r rounds, the maximum score of anyplayer is just r, so the group-score s(g) lies in the range 0 · · · r.The mean group-score taken over all simulations (with the same entry) is denotedS(g) and increases with grade, as shown in Figure 5. The special group withgrade Gc = -2G identified in the previous section, is seen to have a meangroup-score of 2.9. Now in any random-pairing tournament with no byes, theaverage score of all players is given by:
Smid ≡ 12r
2Note that the tiny deviations visible at grades -8G and 4G arise from the rule: a player isnever paired against self.
11
scor
e
grade
0
1
2
3
4
5
6
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 5: Mean group score for tournament Tideal
We call this the mid-score for the tournament, which in the case of Tideal is 3points. It follows that the mean group-score of the special group is very closeto Smid. We define ∆S(g) as the difference of the mean group score from themid-score:
∆S(g) = S(g)− Smid (2)
= s(g)− Smid
The results demonstrate that for Tideal, ∆S(Gc) ≃ 0. Hence the crossing pointsof the two functions ∆G(g) and ∆S(g) are similar i.e. both are nearly equal toGc. We next examine the relation between these two crossing points when thetournament is not ideal.
3.3 Correlation between crossing points for ∆S and ∆G
We can simulate the variety of entries in real tournaments. The number N ofplayers generally increases with number of rounds, so we choose N from a range[Nmin(r), Nmax(r)], where the extremes are linearly increasing functions of thenumber of rounds r. We fix the grade range to be the same as Tideal.
Then, ignoring the complex details for the European player population per grade[3], we generate an entry by choosing the grades for the N players at randomfrom the grade range [-8G, 4G]. Setting
nmin = 5 + 2.5r nmax = 5 + 7.5r
we ensure that most grades are reasonably well populated for all rounds in therange 2 to 10.
We simulate each generated entry as described in section 3.1 to produce rawvalues for ∆G(g) and ∆S(g), for each grade g in the above range. These
12
functions are less smooth than those produced for Tideal, partly due to thereduced sampling rate from 1000 to 100 and partly due to variation in thepopulation per grade. We represent ∆G(g) and ∆S(g) by linear models tosmooth the raw data:
∆G(g) = KG (g − ZG) (3)
∆S(g) = KS (g − ZS)
The slopes K and zero-crossing points Z are obtained from a least squares fitusing 4 points surrounding the changes of sign in the raw data.
The scatter plot in Figure 6 shows that the solutions ZG and ZS are wellcorrelated for all rounds simulated. The correlation coefficient is 0.99, andthere are no obvious outliers.
ZS
ZG
-4.0
-3.0
-2.0
-1.0
0.0
1.0
-4.0 -3.0 -2.0 -1.0 0.0 1.0
Figure 6: Scatter plot for ZS vs ZG in random-pairing
The solution process for a generated tournament Tr-gap is illustrated in Figure7. The rectangle encloses all the points used for fitting the ∆G(g) and ∆S(g)models.
This is a 3 round tournament of 28 players with an irregular entry includinggrade gaps specified in Table 3. The fit produced the coefficients:
ZG = −0.969 KG = −1.024
ZS = −0.496 KS = 0.233
grade -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
entry 1 3 2 1 0 4 0 2 3 3 1 6 2
Table 3: Entry for Tr-gap
Rounding the values of ZG and ZS gives us separated crossing-points at -1 and0 respectively. In comparison, Tideal produces only one crossing point.
13
∆
grade
∆G
∆S
-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 7: Solution process for Tr-gap
3.4 The equilibrium grade
The crossing-point ZG is an equilibrium-grade, where players at that gradeexperience a balanced mix of opponents. There is another equilibrium-gradeat ZS , where players win half their games.
The maximum difference |ZG−ZS | observed in the random-pairing trial is 0.82.This value, less than 1 grade apart, can be expected since the two conditionsare closely related. For if the average grade of a player’s opponents is verydifferent from the player’s own grade, then we would expect the player’s scoreto be different from the average and vice versa.
Since the solution to ∆G(g)=0 is also the point at which ∆G(g)2 is minimised(and similarly for ∆S), we can obtain a unique equilibrium-grade by minimisingthe square sum of the models in Equation (3):
F (g) = K2G(g − ZG)
2 +K2S(g − ZS)
2
The minimisation of F (g) produces the result:
Gc = λGZG + λSZS (4)
λG = K2G/(K
2G +K2
S)
λS = K2S/(K
2G +K2
S)
Thus Gc is a weighted sum of the individual crossing points. Since the weightsare positive and sum to unity, Gc always lies between ZG and ZS .
As we have seen, random-pairing leads to very simple statistics, and indeed thevalue of Gc for each tournament is very similar to the average grade for thetournament. Things are not so simple in Swiss tournaments.
14
4 SWISS PAIRING
4.1 Pairing Method
In the simplest application of swiss-pairing, all players start with zero wins, andpairing in the first round is random. Thereafter, players on the same numberof wins are paired with each other. So the choice of opponents is no longer asfree as it was in the random-pairing method discussed in the previous section.A player’s score is the number of wins, and in a Swiss tournament the averageof all the players’ scores is 1
2r, just as in random-pairing.
In Swiss tournaments, there is the possibility that the pairing matches playerswith different scores when the number of rounds is not exactly log2(n), n beingthe number of players. However, with a reasonable turnout compared to thenumber of rounds, the incidence of these uneven games is normally much lessthan the number of evenly matched games. In the interests of simplicity weignore the problem of uneven matching.
∆
grade
∆G
∆S
-4
-3
-2
-1
0
1
2
3
4
5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 8: Swiss forms for ∆G and ∆S in Tideal
We use a maximum weighted matching algorithm wmatch [5] to carry out thepairing for Swiss tournaments. Following the discussion in [7], the simplestpossible weight assignment for a Swiss tournament gives the weight for a pairingbetween players with scores si and sj as:
Wscore(i, j) = W sech(λ(si − sj)) +W0 (5)
The constant λ controls how severely uneven games are ’punished’. The valueλ = 0.1 was found to be adequate for the simulations presented here. The valuesW = 233 and W0 = 224 were set so that the weighted pairing is also maximumcardinality as discussed in [7]. The locally quadratic form for the weight avoidsthe degeneracy associated with a linear form [8], and ensures that the maximumweight for the pairing is obtained when the players have the same score. Withthe above assumptions, the pairing procedure for a single round is:
15
Algorithm 3. swiss-pairing
SP 1. Shuffle the list of players.
SP 2. For each possible pair ij, choose the weight as in Equation (5)
SP 3. Pair the players using wmatch.
SP 4. Simulate results as discussed in Algorithm simulate-result.
Applying swiss-pairing to Tideal we obtain the forms for ∆G(g) and ∆S(g) shownin Figure 8. Again we see crossing points near to each other at ZG=−2.26 andZS=−1.79, so in swiss-pairing we too have a unique equilibrium grade viaEquation (4).
4.2 Crossing point correlation
The correlation between ZG and ZS for swiss-pairing is similar to that obtainedfor random-pairing. The correlation coefficient is 0.97, again high with noobvious outliers.
ZS
ZG
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0
Figure 9: Scatter diagram for ZS vs ZG in swiss-pairing
The largest difference ∆Z in the crossing points is 1.25, occurring for a 2 roundtournament with 11 players separated into two sections occupying the highestand lowest grades. Apart from this, there are 4 tournaments with ∆Z > 1, thelargest of which has ∆Z = 1.1.
grade -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
entry 1 4 0 4 1 2 0 3 4 2 0 10 1
Table 4: Entry for Ts-gaps
16
This is an 8 round tournament Ts-gaps with 32 players and 3 grade gaps specifiedin Table 4. The raw data for the opponent-grade-difference ∆G(g) shows achange of sign between grades -3G and -1G as seen in Figure 10. So the 4 pointsused for the fit range from -4G to 0G. The mid-score ∆S(g) has raw valueschanging sign between -1G and 0G and uses a shifted set of 4 points from -3G to1G for the least squares fit. The resulting combined solution obtained for Ts-gaps
is Gc = −1.2.
∆
grade
∆G
∆S
-4
-3
-2
-1
0
1
2
3
4
5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 10: Solution process for Ts-gaps
Although there is no concept of a bar in random- or swiss- pairing, it seems thatthe equilibrium-grade takes on one of its roles in the sense that it divides theentry into two parts. Players with grades above Gc win more than half theirgames on average, and so the tournament winner is likely to be found amongstthese players.
The one dimensional models ∆G(g) and ∆S(g) developed for random- andswiss- pairing find a place in mcmahon-pairing, but the introduction of the baradds a whole new dimension to the group behaviour.
5 McMAHON PAIRING
In the ideal McMahon tournament, the entry is well populated with no missinggrades, and Tideal provides us with a good example. Ignoring issues associatedwith the choice of player colour and player club or country, the pairing algorithmcan be reduced to the simple form discussed for swiss-pairing in section 4.1, withthe score there replaced by the mcmahon-score expressed in zero-shodan units.
The bar setting affects the pairing quality [7], and we expose this by simulatingan entire tournament as described in swiss-pairing; then repeat the simulationfor bar settings covering the entire range of grades in the entry.
17
qual
ity
bar
rank-deviation
uneven games
up/dn balance
colour balance
colour alternation
handicap games
0.00
0.05
0.10
0.15
0.20
0.25
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 11: McMahon pairing quality in Tideal
This process applied to Tideal produces the results shown in Figure 11. Eachquality item is normalised so that it has a range from best quality (0) to worstquality (1). The quality components shown are:
uneven games The number of uneven games for all players.
up/dn balance The magnitude of the difference between the number of gamesplayed up and games played down.
handicap games The number of games where handicap stones are awarded -i.e. when the mcmahon-score difference for a pair exceeds 1 point.
colour balance The magnitude of the difference between games played whiteand games played black.
colour alternation The magnitude of the number of colour reversals relativeto the expected colour reversals.
rank-deviation The root-mean-square of the difference between the player’smcmahon-rank and grade-rank.
The only active weight in the pairing algorithm is the McMahon weight ensuringas many even games as possible, and this has its greatest direct effect on thefirst three qualities only.
We note firstly that there is a low incidence of handicap games for all bars below4G. There are a total of 156 games in the tournament, and the worst handicapquality of 0.042 translates to a mean of 6.6 handicap games per tournament.For bars from -1G and above, the uneven quality lies in the range 0.12 to 0.14,and this would produce a mean of 3.2 to 3.8 uneven games per round.
At the end of the tournament, players are ranked by mcmahon-score scaled tolie in the range [0, 1]. We also rank players by grade, also scaled to lie in the
18
same range. We call these the mcmahon-rank and grade-rank respectively. Therank deviation shown in Figure 11 is the root-mean-square difference of the tworanks for each player. The rank deviation improves by a factor of two as thebar increases.
The quality analysis confirms that the simple pairing algorithm employed copeswell with the uniform entry of Tideal at all bar settings. The introduction of thebar is an independent variable affecting many characteristics of the McMahonand we take this up in the next sections.
6 TOP GROUP PEER GAMES
The bar can be set at any grade whatsoever, but the top groups will not bewell served if it is set too high or too low. The top groups may include allthe grades needed to ensure that the population exceeds the number of rounds.In the extreme case of a long tournament, where the entry is very thin at thehigher grades, this could mean a huge grade range in the top groups. In such asituation, not all the players in these groups could be considered as peers.
peer
gam
e ra
tio
bar
-6G-5G-4G-3G-2G-1G0G1G2G3G4GΨ(b)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 12: Peer Games
Define a top-group Q(g) to consist of all players with grades from g to gmax. Wethen examine how the bar setting b affects the proportion of peer-games in thistop-group.
Suppose there are a total of Ng games played in the top-group Q(g). Thenumber of games between players in the top-group depends on the bar, and isdenoted by Eg(b). Such games are the peer-games, and we are interested in thebehaviour of the ratio of peer-games to Ng for each grade g, as we vary the bar.
Ψg(b) = Eg(b)/Ng
19
The graphs in Figure 12 show the behaviour of the peer-games-ratio Ψg(b) asthe bar is varied. In particular we observe that for each grade g, Ψg(b) risessteadily as the bar b is increased from its lowest value at gmin. However, Ψg(b)then abruptly levels off for bar values from b = g to b = gmax.
Consider the topmost group at 4G i.e. Q(4). If the bar is set to gmin, the topmostgroup has a peer-games-ratio of 18.3%, so only 4.4 games out of the maximumof 24 is a peer-game. If the bar is set to gmax then the ratio rises to 49.2% -which is almost as high as it can be. At this bar, the 4 top players then do getto play each other.
The behaviour for the top group Q(3) is quite different. Again at a bar -8G thepeer-games-ratio is low. However, at a bar grade of 4G the ratio is a healthy81.3%, and the bar can be reduced to 1G whilst still keeping the peer-games-ratioabove 50%.
The bar-group consists of all players with grades in the range b to gmax whereb is the bar setting. The peer-games-ratio for the bar-group is given by Ψb(b).This is illustrated by the (dotted) graph Ψ(b) = Ψb(b) shown in Figure 12. Itis clear that for bars below the maximum grade, the peer-games-ratio for thebar-group is not very sensitive to the bar setting.
7 OPPONENT GRADES
The McMahon system pairs players on the same mcmahon-score, so if grades arefairly populated, we can expect that players well below the bar play others onsimilar grades. Above the bar, players may meet opponents of widely differinggrades, especially in the early rounds. But this grade difference should narrowas the stronger players win through in the later rounds.
The average grade of opponents of players in group g depends on the bar b as wellas on g, for the bar setting influences whom you get to play. We can extend thegroup opponent grade notion developed for random-pairing and swiss-pairing.Define Gg(b) to be the average of the grades of opponents of group g, when thebar is set to b.
The incidence of equally graded games in McMahon tournaments is indicatedsimilarly to (1), by the mean opponent-grade-difference ∆Gg defined by:
∆Gg(b) = Gg(b)− g (6)
The behaviour of ∆Gg(b) for the weakest group in Tideal is shown highlightedin Figure 13. We see that ∆G settles down to a value of about 0.7 for all barsbeyond -3G. As expected, the weakest players in the tournament are playingagainst stronger players, no matter where the bar is set (i.e. ∆Ggmin(b) > 0).The strongest group 4G in Figure 13, also highlighted, shows a grade differencesteadily reducing in size as the bar increases from the weakest to the strongestgrade. The strongest group always has opponents the same grade or weaker, so∆Ggmax(b) < 0 for every bar.
20
∆G
bar
-8G-7G-6G-5G-4G-3G-2G-1G0G1G2G3G4G
-4
-3
-2
-1
0
1
2
3
4
5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 13: McMahon opponent grade difference
For the extreme strongest and weakest grades, we can see that the variation ofopponent-grade-difference is very smooth. This however is not the case for anygrade in between. Indeed we observe that for every grade between the weakestand the strongest, the group g has a peak value for ∆Gg when the bar b= g.To put this another way, set the bar at b somewhere in the range gmin+1 togmax−1. Then ∆Gg(b) < ∆Gb(b) for all grades g other than gmin or gmax.
These results make it clear that the group at the bottom of the bar enjoys aspecial status, and really does stand between the weaker and stronger playersin the tournament. We saw a similar phenomenon in the discussion on peergames, where the peer-game-ratio Ψg(b) undergoes a sharp change when b=g.
In the (grade, bar) plane, the line b= g provides a layer separating the weakerfrom the stronger players, and we call this the bar-layer. Define the opponentgrade difference on the bar-layer:
∆G(b) = ∆Gb(b) (7)
We saw earlier that ∆G(gmin) > 0 and that ∆G(gmax) < 0. Therefore there is azero crossing point for ∆G(b), somewhere between the minimum and maximumgrades.
The graph in Figure 14 shows the values obtained for ∆G(b) extracted fromthe peak values of the individual grade graphs in Figure 13. The graph clearlyshows a crossing point in the neighbourhood of 3G. This is an equilibrium-gradeanalogous to the one found for random-pairing and swiss-pairing.
When the bar is set to the equilibrium-grade players at the bottom of the barmeet roughly equal numbers of stronger or weaker players. At this bar settingthere are also a healthy number of peer games as discussed in Section 6. Theequilibrium-grade is a unique grade in McMahon tournaments just as it was inRandom or Swiss tournaments.
21
grad
e di
ffere
nce
bar
∆G(b)
-1
0
1
2
3
4
5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 14: ∆G on the bar-layer
8 GROUP SCORES
We have so far discussed the influence of the bar on the choice of opponents,and now we consider how the bar setting affects the score of players. Extendingthe swiss-pairing discussion, we define the group-score s(g, b) as the average ofthe mcmahon-scores in each group g when the bar is set to b.
We simulate the tournament Tideal to produce the mean group-score:
Sg(b) = s(g, b) (8)
for each grade g and bar b. The graphs in Figure 15 show that Sg(b) levels off toa stable value Slim(g) as the bar rises, for all grades g except gmax. The stablevalue for each grade can be defined as the value at the maximum possible barsetting:
Slim(g) = Sg(b=gmax) (9)
We observe that Slim(g) changes in a regular manner: values are spaced verynearly 1 point apart at every grade. Moreover for g = 0, Slim(g) = 2.96, verynearly a perfect average score for the shodan group in a 6 round tournament.This means that to a good approximation the stable score can be representedby Slim(g)=g + 1
2r
In Section 7, we saw that when the bar is set to the maximum possible, any groupg sees a balanced mix of opponent grades for gmin < g < gmax. We can thenexpect that most players will win roughly half their games. So players in group gwill end up on the same final mcmahon-score which is the initial-mcmahon-scoreplus half the number of rounds3 r.
3In the ’zero shodan’ grading representation, the initial-mcmahon-score of a player is thegrade of the player (or the bar grade) in zero-shodan units. Other systems may have a differentorigin for the mcmahon-score.
22
This is a universal mid-score for McMahon tournaments defined along the samelines as in random-pairing or swiss-pairing for each grade g by :
Smid(g) = g + 12r (10)
grou
p-sc
ore
bar
-8G-7G-6G-5G-4G-3G-2G-1G0G1G2G3G4G
-6
-4
-2
0
2
4
6
8
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 15: Mean group-score for McMahon
Therefore Slim(g) ≃ Smid(g) for all grades apart from the highest and lowest.We present the difference as the graph labelled Slim −Smid in Figure 16. Whenthe bar is set to a value lower than gmax the group-score of every group above thebar is strongly affected, and for most groups it no longer tracks the mid-scorevalue i.e. Sg(b) = Smid(g) when b = gmax.
grou
p-sc
ore
scor
e-di
ffere
nce
bar
Smid
SSlim - Smid
∆S
-8
-6
-4
-2
0
2
4
6
8
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Figure 16: Group-score and differences on the bar-layer
23
Define the difference:∆Sg(b) = Sg(b)− Smid(g)
The behaviour of the group-score for players at the bottom of the bar i.e. onthe bar-layer identified in the previous section is expressed by Sb(b). We setg=b in Equation (8) and define:
S(b) = Sb(b) (11)
∆S(b) = ∆Sb(b)
= S(b)− Smid(b) (12)It is apparent from Figure 16 that S(b) < Smid(b) up to the bar b = 3G. Beyondthat point S(b) > Smid(b). Hence for our ideal tournament, we have the resultthat ∆S(b) changes sign near the value b = 3G.
This gives the same zero-crossing point as we found in the discussion of opponentgrades in Section 7. The feature that we found in both random-pairing andswiss-pairing, namely that ∆G and ∆S have similar zero crossing points, nowemerges in mcmahon-pairing for players in the bar-layer.
9 SCORE DISTRIBUTIONS AND THE BAR
9.1 Player score and Group score
A player’s individual score is determined both by the probability of win andopponent choice. The group-score is determined by probability of win and thechoice of all opponents in the group. Since all n players in the group have thesame grade, they have the same score range [smin, smax].
The player-score distributions in a given group are not necessarily independent,since the sum of the scores in a pair is 1. Nevertheless, since the players inthe group have the same grade, we can assume that they have identical scoredistributions denoted by p(si), where si is the score of the ith player in thegroup.
The group-score s is proportional to the sum of the individual player scores:
s = k
n∑i=1
si
Here n is the number of players in the group, and k = 1/n. Since the mean ofa sum of random variables is the sum of the individual means [9], and since theindividual means are the same, it follows that the mean player score is the sameas the mean group-score.
The variances of the group score and the player score are however very different.For example, Figure 17 shows probability distributions for the shodan groupwhen the bar is set to 0G. The player-score distribution is much broader than
24
prob
abili
ty
score
player-scoregroup-score
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6
Figure 17: Shodan player-score and group-score histograms
the group-score distribution and its variance is 4.46 times the variance of thegroup-score distribution.
This result confirms that individual player scores are not independent. For ifthey were, the player-score variance would be exactly 4 times the group-scorevariance [9], since there are 4 players in the group. The excess shows that playerscores in the group are positively correlated.
For the purpose of separating players above the bar in McMahon tournaments,the group-score provides a sharper instrument than player-score.
9.2 Winning chances above the bar
One of the commonly stated guidelines for McMahon tournaments is that playersabove the bar should have a reasonable chance of winning. It sometimes happensthat the tournament winner does not win every game, but will win more gamesthan any other player above the bar. Hence the probability that a group scoresin excess of a particular value may help to quantify winning chances.
More precisely, we are concerned with the probability T (g, b, s) that a group gscores more than s when the bar is set to b. Formally:
T (g, b, s) = Prob(group-score > s | grade=g, bar=b) (13)
This probability is the complement of the cumulative distribution function, andis known as the tail distribution [10]. The tail distributions in Tideal for highergrades at a bar setting of 0G is shown in Figure 18. The vertical line in theplot at group-score 3 identifies the mid-score Smid(0), and its length covers theprobability range [0.05, 0.95]. The highlighted graph is the shodan group at thebottom of the bar. It is apparent that the probability that it scores more thanthe mid-score is very low, and indeed it has no chance at all of winning.
25
prob
abili
ty
group-score
-1G
0G
1G
2G
3G
4G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6
Figure 18: Tail distributions T (g, 0, s) for bar at shodan
All groups below shodan have no chance of scoring more than Smid(0). Thegroups 3G and 4G above shodan have some chance of scoring more than 5 andalso have a very good chance of scoring more than Smid(0). For reference, the4G individual probability of scoring 6 is just 10.2%.
There is no acceptable value to be obtained for a ’reasonable chance’ of winning.We can say however that the tail distribution provides a clear separation betweengrades: 0G and below have no chance of winning; 3G and above have some chanceof winning.
9.3 Bar separation
The bar-group should clearly separate the performance of players above the barfrom the performance of players below the bar.
In the tournament Tideal we observe in Figure 18 that when the bar is set to0G, T (g, 0, Smid(0)) is very small for grades below grade 1G. This bar providesno distinction between -1G and 0G groups.
By setting the bar at a higher value, we might be able to improve the differencein performance of players at the bottom of the bar, from those below the bar.The tail distribution for each grade g at the mid-score for each bar b is obtainedfrom Equation (13) with the group-score s set to Smid(b). This gives us therestricted form Tmid(g, b):
Tmid(g, b) = T (g, b, Smid(b)) (14)
The mid-score performance for the shodan group is illustrated in the highlightedgraph in Figure 19. Clearly we need to increase the bar to beyond the 2G settingto see any significant increase in performance of players at the bottom of thebar i.e. those in the bar-layer.
26
prob
abili
ty
bar
-4G
-3G
-2G
-1G
0G
1G
2G
3G
4G
∆T(b)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 19: The mid-score tail distributions Tmid(g, b)
The mid-score performance of the players in the bar-layer is given by Tmid(b, b)and illustrated by the dotted graph in Figure 19. The relation between the singlevariable Tmid(b, b) and its two-variable parent Tmid(g, b) for Tideal is shown inFigure 20.
Tmid(g, b)
Tmid(b, b)
Tmid(b-1, b)
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
bar-8
-6
-4
-2
0
2
4
grade
0.00.20.40.60.81.0
Tmid
Figure 20: Mid score tail distribution Tmid(g, b)
The mid-score performance of players just below the bar is given by Tmid(b, b-1),and is illustrated by the horizontal line in Figure 20. In Tideal, the performancejust below the bar is zero for all bars, but this is not always the case fortournaments with a more varied entry.
27
We formally define the bar-separation ∆T (b) as the difference in the performanceacross the bar-layer:
∆T (b) = γB(b)− γL(b) (15)
γL(b) = Tmid(b− 1, b)
γB(b) = Tmid(b, b)
The bar-separation ∆T (b) can be used as a means for judging the bar setting,independently of how it was arrived at. Thus, if ∆T (b) > 0 it is good otherwisethe setting may need further investigation. In Section 7 and Section 8 we sawthat both ∆G and ∆S had similar crossing points near grade 3G. A bar at 3Gis positioned well up the the curve ∆T (b) , and so is judged to be good.
For non-ideal tournaments, the shape of ∆T (b) is much sharper than illustratedin Figure 19. It is usefully modelled by a ramp [11] function, discussed inAppendix A.
10 BAR DETERMINATION METHOD
The functions ∆G in Section 7 and ∆S in Section 8 provide us with the maintools for determining the bar from the tournament’s entry. We firstly present amethod for doing this, and then specify a Monte Carlo simulation trial to testthe assumptions made in the method and to explore round-dependent features.
10.1 Statement of the method
We are given a tournament with a specified entry and number of rounds.
Algorithm 4. find-bar
B1. Set the bar at the highest grade.
B2. Simulate the tournament for Np pairings.
B3. Collect the mean group-score and mean opponent-grade for the group atthe bottom of the current bar.
B4. Lower the bar to the next non-empty group.
B5. Continue at B2 for 4 successive bars.
B6. Fit linear models to the mean group-score and mean opponent-grade data.
B7. Apply Equation (16) to obtain the bar setting.
In step B6, we fit linear models described in section 3.4 to find the zero crossingpoints and equilibrium grade. This step always succeeds. If there is just onegroup in the tournament it falls back to Swiss and there is no bar. This is
28
equivalent to setting the bar in a McMahon tournament to gmin. If there aretwo or more groups, then there is always a solution, because the variance of thesequence of bar values is non-zero.
In the above linear models, a higher value of K implies a more reliable crossingpoint. As we have seen in the graphs of ∆G in Figure 13 and ∆S in Figure 15,the zero crossing points are similar, but not identical. We take account of thecrossing point reliability in step B7, by minimising the square sum of the linearmodels as in (4) to produce a weighted solution bsol which always lies betweenthe above crossing points:
bsol =K2
GZG +K2SZS
K2G +K2
S
(16)
The actual bar grade Bsol is obtained as the nearest non-empty grade to bsol.
The simulation time for a single pairing increases as n3, so in step B2 we need tokeep the entry in the simulated tournaments to the minimum possible. Howeverwe do not need to simulate the entire entry in a tournament. We certainly needthe top 4 groups and all the groups below the top 4 which might possiblyinfluence the performance of the top players. We thus only need to simulate thetop 4 + r groups at most.
We have in effect seen the successful application of the above algorithm to thecase of Tideal.
10.2 Scope of the Monte Carlo trial
We specify a large-scale trial whose purpose is:
� Expose any entries preventing solution.
� Identify outliers.
� Identify features exhibiting clear dependency on the number of rounds.
� Generate a bar population table to compare with the traditional guidelines.
10.3 Solution failure
Some particular entries could prevent the above algorithm from producing anacceptable bar. Possible conditions for solution failure are:
There may be a flawed pairing. This would cause step B2 to fail. A fewsuch failures in repeated trials of the same entry would not be important,but a high rate of failure would mean that the entry is not capable ofsupporting a McMahon tournament.
Unacceptable solutions. For example, if the models for ∆G(b) and ∆S(b)have weak slope parameters, the solutions may be very far apart and thesquare sum may have a minimum which lies outside the range of gradesin the tournament.
29
10.4 Variation with number of rounds
We now have the means to generate a classic table relating the bar populationto the number of rounds. There are a number of other important features thatmay show a trend with number of rounds. One of these is the bar-depth definedas the difference between gmax and Bsol. Others include:
Solutions of ZG and ZS may differ. In Tideal we saw that the solutions weresimilar. We collect the coefficient for the correlation between zero crossingpoints.
Bar separation. We monitor the ramp hinge and slope parameters for ∆T .
Bar population. We obtain the population distribution and generate a bartable from the mean bar population as a function of the number of rounds.
Quality. As already mentioned, it is difficult to set the bar giving a populationwhich guarantees a unique winner . We monitor the values for probabilityof a unique winner, as well as rank-deviation.
10.5 Tournament entry
The trial generates tournaments for rounds ranging sequentially from 2 to 10.For each number of rounds r, we first choose a total even entry n at randomfrom a range [nmin, nmax] depending on r:
nmin = Nmin + rRmin
nmax = Nmax + rRmax
The variation in limits helps to provide a reasonably full population for therelevant higher grades in the tournament, but also allows some grades to havea zero entry. Once the total entry is known, we then assign to each of the nplayers a grade chosen at random from the range [gmin, gmax].
The limits for the entry generation are set to the values in Table 5.
Nmin Rmin Nmax Rmax gmin gmax
value 20 4 25 5 -8G 4G
Table 5: Limits for the tournament entry
For a 10 round tournament, the maximum number of players is 75. A -8G playercould affect a bar set at 2G by winning all games (but this is virtually impossibleas we have seen). For a two round event, the minimum population is 28 andthe -8G player would affect a bar set at the very unlikely grade -6G.
Once an entry is generated as described above, we find the bar according toAlgorithm find-bar. Then repeat the process for 200 different entry samples,all with the same number of rounds.
30
10.6 Pairing sample rate
For each bar setting scanned by Algorithm find-bar, we need to simulate anumber of pairings. All the results obtained for Tideal are based on a sampleof 1000 pairings. A sample rate lower than 100 can produce some severedistortions. For example, consider the 5 round tournament Tm-gap with entry:
grade -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
entry 6 0 5 4 0 5 4 6 4 0 3 4 1
Table 6: Tm-gap entry 5 rounds 42 players
The leftmost plot in Figure 21 shows the ∆G and ∆S values for 10 pairings ofTm-gap. The variations in the values are significantly reduced on simulating thetournament with 100 pairings as seen in the centre plot of Figure 21. Increasingthe numbers of pairings beyond 1000 does not yield visible improvement.
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4
prob
abili
ty
bar
γB
γL
ΓB
ΓL
∆T
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-5 -4 -3 -2 -1 0 1 2 3 4
Figure 21: Sampling at 10 and 100 pairings for Tm-gap
The rightmost plot shows values for the bar separation function ∆T discussedin section 9.3. A model for the bar-separation is presented in Appendix A. Theterm ΓB(b) is the dominant component of the model for ∆T and this is shownin the rightmost plot of Figure 21.
Values for the derived parameters required in the bar solution and bar separationat increasing numbers of pairings are shown in Table 7
With 100 pairings per tournament the accuracy of bsol at (±0.1) is sufficientfor our purposes. The steady state values for HB and KB are reached at 2000pairings, beyond which the change is within (±0.02)
31
pairings bsol HB KB
10 2.270 0.833 0.400100 2.149 1.364 0.3301000 2.126 1.465 0.3382000 2.132 1.548 0.354
Table 7: Parameter accuracy dependent on number of pairings
10.7 Software
Software written to produce all the results in this document is called toursim
[12]. It is a Linux command line application released under the GNU GeneralPublic License.
This program is a work in progress: there is no manual but the source iscommented. The output files [13] required for results in this document containextensive supporting detail.
11 MONTE-CARLO TRIAL RESULTS
11.1 Solution failures
There are 1800 different tournaments simulated over the entire round range and100 complete pairings per entry generated. There are no flawed pairings in the180,000 pairings sampled.
The solution bsol lies above the maximum grade in 4 tournaments (all 2 rounds),but in all cases by less than 0.6 grade. In all these cases the maximum grade hasan entry of 4 or more. Since there is an absolute limit on the bar population of2r players, there would be no need in practice to invoke Algorithm find-bar.The algorithm nevertheless does produce the value Bsol = gmax in these cases.
11.2 Solution correlation
The solution pair (ZG, ZS) for every entry in the trial is presented in the scatterplot Figure 22. The three points A,B,C identify clear outliers. The rectanglecontains the 2-round tournaments discussed in the previous section.
point rounds ZS − ZG -1G 0G 1G 2G 3G 4G bar
A 7 0.90 6 7 1 0 1 6 3GB 4 1.14 5 0 0 2 1 4 3GC 4 1.37 6 6 3 0 1 5 4G
Table 8: Outlier features
32
ZS
ZG
A
B
C
1.0
2.0
3.0
4.0
5.0
6.0
1.0 2.0 3.0 4.0 5.0 6.0
Figure 22: Scatter plot for ZS vs ZG in mcmahon-pairing
One common feature of the models for the outliers seen in Figure 23 is that thatthe fit to ∆G(b) and ∆S(b) seems to be poor. Even if we were to interpolatethe solution for the crossing points directly from the raw data, we still obtainestimates for Bsol which agree with Algorithm find-bar.
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
-1 0 1 2 3 4
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
-1 0 1 2 3 4
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
-1 0 1 2 3 4
Figure 23: The Toutlier models
Simple interpolation may fail however if the changes in sign for ∆G(b) and∆S(b) lie at the ends a large grade gap. The entries for the higher grades inthe outliers are given in Table 8, and they all show grade gaps 1 or more gradesbelow the solution point.
11.3 Bar Depth
For any number of rounds, the observed bar-depth lies in the range 0 · · · 4. Theleft hand plot in Figure 24 shows histograms for the bar-depth distributions.
33
prob
abili
ty
bar depth
2R3R4R5R6R7R8R9R10R
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-1 0 1 2 3 4
bar
dept
h
rounds
mean
sigma
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
2 3 4 5 6 7 8 9 10
Figure 24: Bar depth statistics
The individual plots are displaced along the bar-depth axis to distinguish thedifferent number of rounds.
For each number of rounds, the most common bar-depth is 1 grade, i.e. the baris just 1 grade below the highest grade.
There is 1 case with a bar-depth of 4. This is a 6 round tournament, where thetop five grades from 0G to 4G have entries 7,0,0,2,3, and the bar obtained is atgrade 0G.
It is very rare for the bar-depth to be as high as 3. In almost all cases whenthis happens, there is a gap in the entry at the higher grades. When thereis a missing grade say gmiss near gmax but nevertheless a healthy populationabove gmiss, the bar sometimes does not cross the gap. This is illustrated inthe tournament Tm-gap discussed in Section 10.6. It is a 5 round event withBsol = 2 giving a population of 8 players. There is a gap at 1G and then a groupof 4 players at 0G. Crossing the gap would induce large changes in the values of∆G and ∆S, moving the solution well away from optimal.
There is one case where the bar-depth is 3, and there are no missing grades.This is also a 6 round tournament with top 5 grades 7,5,2,1,1. Although thereare no missing grades, the top groups are very thin and the algorithm lowersthe bar to 1G giving a bar population of 9.
The right hand plot in Figure 24 shows that the mean bar-depth increases withthe number of rounds, but the rate of change per round is quite small averaging0.09 per round.
11.4 Bar Separation
As mentioned in section 10.5, we smooth the individual components of ∆T (b)via the ramp models ΓB(b) and ΓL(b) discussed in Appendix A. Figure 25 shows
34
grade -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
entry 7 3 4 5 5 4 2 3 3 9 1 3 1
Table 9: Tmcmahon entry 7 rounds 50 players
the models arising from a 7 round tournament Tmcmahon and Table 9 shows theentry generated in the trial.
The left hand plot indicates a bar solution Bsol = 2G. The right hand plotconfirms that this bar is well up on the ramp ΓB.
The BGA tables in Section 1 suggest a maximum bar population of 18 playersfor a 7 round tournament, and Table 9 would then allow a bar at 0G. At thisgrade ∆T is zero, meaning that there is no distinction in the performance of 0Gfrom -1G players.
∆
bar
∆G(b)
∆S(b)
-2
-1
0
1
2
-2 -1 0 1 2 3 4
prob
abili
ty
bar
γB
γL
ΓB
ΓL
∆T
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-2 -1 0 1 2 3 4
Figure 25: Solution process and ramp models for Tmcmahon
We are interested in the probability Psep that the bar-separation is non-zero atthe bar b=Bsol. The slowly decreasing values in Figure 26 are consistent withthe observed increase in bar-depth for increasing population found in section11.3. For players at the bottom of a deep bar will meet an increased number ofstronger opponents and so find it harder to beat the mid-score.
The dominant component in ∆T is the ramp function ΓB with hinge point HB ,and it is of interest to see how far up this ramp the bar solution rises. To thisend we consider a normalised value:
Θ = (bsol −HB)/(gmax −HB)
The value of Θ lies in the range [0, 1] when bsol is on the ramp, and becomesnegative when it is below the ramp’s hinge. The higher the value of Θ, thebetter is the separation in the performance of players at the bottom of the bar,compared to those below the bar.
35
ram
p st
ats
rounds
Psep
Θµ
Kµ
0.0
0.2
0.4
0.6
0.8
1.0
2 3 4 5 6 7 8 9 10
Figure 26: Ramp statistics
The slope KB in the ramp model has a maximum value of 1 corresponding tothe case where the bar is at the maximum grade and the value of ΓB is zeroat grade gmax − 1. The quantities Θ and KB all have the same range and arepresented along with Psep in Figure 26. The mean normalised separation Θµ
shows a decreasing trend with number of rounds. This means that bsol is gettingcloser to the hinge point of the ramp, and this is consistent with the decreasein Psep observed above.
11.5 Bar Population
prob
abili
ty
bar population
2R3R4R5R6R7R8R9R10R
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 2 4 6 8 10 12 14 16 18 20
Figure 27: Cumulative distribution for the bar population
The underlying distribution for the bar population shown in Figure 27 revealsa fairly regular dependence on the number of rounds.This regularity emerges in the bar table, which here we present graphicallyin Figure 28. The dotted graphs display the BGA minimum and maximumpopulation [14] range. The actual obtained minimum in the trial is somewhat
36
popu
latio
n
rounds
max
mean
median
min
0
2
4
6
8
10
12
14
16
18
20
22
2 3 4 5 6 7 8 9 10
Figure 28: Bar table statistics
below the classical minimum, defined as one more than the number of rounds.The cdf plot in Figure 27 indicates for example, that in a 6 round tournament,there is a 14% chance that the bar population is lower than the number ofrounds.
Note that the trends observed in this trial are to some extent associated withthe assumptions made for the dependence of total entry size on the numberof rounds, as defined in Table 5. The rough nature of the trial results forminimum and maximum populations reflects the low number (200) of differenttournaments sampled for each round, but the trends are clear.
11.6 Uniqueness and ranking
It is desired that a tournament produces a unique winner without recourse totie-breaks. There is also a desire that the tournament should produce a sensibleranking for runners up. These requirements are often incompatible.
In Figure 29 we present a normalised version of the rank deviation illustratedin Figure 11, along with the probability of a unique winner for all the possiblebar settings in the tournament Tm-gap.
The normalised grade-rank RG and mcmahon-rank RS have values lying in therange [0, 1] and are defined by:
RG(g) = (g − gmin)/(gmax − gmin) (17)
RS(g, b) = (s(g, b)− Smin)/(Smax − Smin)
In zero-shodan units Smin = gmin, and Smax = b + wmax, where wmax is themaximum number of wins above the bar. The player’s mcmahon-score is givenby s(g, b).
The graph labelled δrank shows the values of the rms-difference between RG(g)and RS(g, b) summed over all grades and 1000 pairings in the simulation of
37
bar
Punique
δrank
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Figure 29: Unique winner probability and rank deviation for Tm-gap
Tm-gap. It decreases slowly with increasing bar, and the behaviour seen here istypical.
The graph Punique shows a modest peak at bar 3G, but the behaviour near thebar solution takes on many different forms for different entries. Tm-gap providesa unique winner at a bar grade -5G. Referring to Table 6, this gives a populationof 31. A player is drawn up at round 1 to give a perfect population of 32 for a5 round tournament. Such a bar is of course out of the question!
12 GUIDELINES
In the light of the results obtained in this study, we revisit the guidelines forsetting the bar.
12.1 Classical guidelines and maxims
Where possible, we supply additional information to clarify the guideline.
Bar set too high. Top players run out of peer opponents early. For Tideal theproportion of peer games decreases slowly as the bar is lowered from themaximum (Section 6).
Bar set too low. The top players may never meet. For Tideal the proportionof peer games played by the topmost group drops from 50% to 26% whenthe bar drops to shodan.
Smallest bar population It is generally accepted that the population abovethe bar should be at least the same as the number of rounds. Howeverthe bar population distribution obtained in the Monte Carlo trial does notexclude values lower than the number of rounds.
38
Winning chances above the bar. The winning probability above the bar isvery sensitive to grade, and players certainly do not have an equal chanceof winning. There is no natural way of defining a ’reasonable’ chance towin.
Unique winner For tournaments of 4 rounds or more, the simulations carriedout in this study showed that on average, the probability of a uniquewinner is less than 60%
12.2 Additional guidelines
The emphasis throughout has been that the position of the bar is sensitive tothe actual number of players in each grade group. There is no simple formulafor setting the bar given the entry distribution, and in the end we require analgorithm such as the one discussed in Section 10 to produce a solution. Somesimple rules of thumb have emerged though, and these might be useful whensetting the bar manually.
Avoid the highest grade. It is unusual for the bar to be set at the highestgrade for tournaments of more than 4 rounds. In the Monte Carlo trial,the probability of a bar at the maximum grade is less than 2%.
Use the highest grade where needed. For the smaller tournaments in therange 2 to 4 rounds, the probability of a bar at the top grade ranges from10% to 30%. This may happen when there is a relatively large top groupfollowed by a gap in the entry.
Bar depth. For most tournaments, the bar-depth should be 1 or 2. It is veryunusual for the bar-depth to be 3, and the most common bar-depth is 1for all number of rounds.
Mind the gap. Where there is a grade with zero or very few players within4 grades of the maximum grade, be careful of crossing the gap to boostthe bar population. This can lead to an excessive bar-depth and poorperformance for players at the bottom of the bar. The tournament Tm-gap
is a case in point. It might be tempting to cross the gap at 1G and set thebar at 0G to give the populations of 12 according to Table 6. The graphsin the centre plot of Figure 21 show just how far the performance of theplayers at the bottom of the bar would move from the ideal.
39
13 SUMMARY AND CONCLUSION
13.1 Assumptions
The key assumptions made in this study are:
Strength. A player’s grade defines the exact strength of the player used insimulating game results.
No repeats. There are no repeat games, irrespective of the pairing method.
Pairing. Apart from random-pairing, the pairing algorithm applies a maximumweight to players with equal scores.
Entry distribution. The number of players in a tournament is even, andwe assume that the available pool of players increases linearly with thenumber of rounds. Player grades are chosen at random from a fixed range.
Winning probability. The probability of win between players of differingstrength is derived from the E.G.D published statistics and is definedin Appendix B.
13.2 Key quantities
The study has identified key quantities applicable to any pairing method.
Mid score. In any single tournament, the mid-score for random- and swiss-pairing is half the number of rounds. For mcmahon-pairing it depends onthe group grade, and is the initial mcmahon-score plus half the number ofrounds.
Group score. The average of the scores of all players in the group.
Group opponent-grade. The average grade of all opponents of players in thegroup.
Mid Tail distribution. Probability that the mean group-score exceeds themid-score.
Rank deviation. Each player is assigned a grade rank and a score rank asdefined in Equation (17). The rank-deviation is the root mean squaredifference in the two rankings.
40
13.3 Conclusion
The main conclusions to be drawn from this study are:
Algorithm. A Monte Carlo algorithm for determining an equilibrium-gradefrom the player entry grades in Random, Swiss, or McMahon tournamentshas been identified. It relies on minimising a quadratic form constructedfrom the mid-score, group-score, and opponent-grade.
Sensitivity. The equilibrium-grade is very sensitive to the exact distributionof player numbers in the top grades.
No formula. No simple formula in Swiss or McMahon tournaments has beenfound which relates the equilibrium-grade to the number of rounds orentry statistics, such as the first 4 moments or entropy of the populationsper grade.
Unique winner. The probability of a unique winner is a function of the bar.No feature of this function has been found which is directly related to theequilibrium grade for arbitrary entries.
Rank deviation. The rank-deviation is a function of the bar. This functiongenerally rises for low bar settings and then falls gently as the bar isincreased to the maximum grade.
The bar. The equilibrium-grade is a prime candidate for setting the bar in anyMcMahon tournament.
41
A MODEL FOR BAR SEPARATION
The behaviour of Tmid(b, g) along the bar-layer line g = b as illustrated in Figure19, consists of a long horizontal flat section followed by a rapid and fairly linearrise. This behaviour can usefully be modelled by a ramp-function. The rampturns at a point HB called the hinge, and can be expressed in the form:
ΓB(b) = KB(b−HB) for HB ≤ b ≤ gmax (18)
= 0 for b < HB
For our purposes the sloping part can be obtained by a least squares fit toγB(b) ≡ Tmid(b, b) values for the 3 highest non-empty grades.
The mid-score performance for players just below the bar can also be representedby a ramp function. In this case the hinge is usually at a much higher gradethan for the bar-layer so the ramp part is very short. The ramp is obtained bya least squares fit to the top three non-empty grades for γL(b) ≡ Tmid(b, b− 1).
If there are missing grades in the entry near gmax, then there are some pointswhere the raw data γB(b) is not defined. Such points are ignored in the leastsquares process, and we simply search for the three largest grades with non-zeropopulations.
Missing grades affect the data for γL(b) more severely, since even if the entry atgrade b is non-zero, there might be a gap at b − 1. Again, for each non-emptygrade b, we search for the highest non-empty grade below b. This grade can bedenoted by b ⊖ 1. So the raw data required for modelling the performance ofplayers below the bar takes the form:
γL(b) = Tmid(b, b⊖ 1), n(b) > 0
This raw data leads to a ramp model ΓL(b) having a hinge at HL, and slopeKL.
The model for the bar-separation ∆T (b) function is defined as the difference inthe ramp models across the bar-layer:
∆T (b) = ΓB(b)− ΓL(b) (19)
B PROBABILITY OF WIN
A model [2] for the probability of win between players with ratings r and s isbased on published data provided by the European Go Database. The modelcan be expressed using the standard S-shaped error function erf as follows:
p(r, s) = 12 [1− erf(Λ(r, s))]
Λ(r, s) =
3∑n=0
hn(s− r)enK min(r,s)
hn(x) = unx+ vnx3, n = 0 · · · 3
42
Here un and vn are positive constants specifying monotonic increasing cubics,and the best-fit non-zero values for the coefficients are:
u0 v0 u1 u3 K
0.0351224 0.00445376 0.156777 0.0164481 0.18818
Table 10: Coefficients for winning probability
C GLOSSARY
all-play-all(p7): Each player has one game against each of the others.
bar-depth(p30): The difference between the bar grade and the maximumgrade.
bar-group(p20): The collection of players with grades ≥ bar.
bar-layer(p21): A line in the (bar,grade) plane representing players at thebottom of a variable bar.
BGA(p35): British Go Association.
crossing-point(p11): A sequence f(i) changes sign between i and i+1. Inverseinterpolation provides the crossing point. .
edges(p7): An edge in a graph contains exactly two distinct vertices.
E.G.D(p6): European Go Database.
equilibrium-grade(p14): A grade minimising the square sum of ∆S and ∆G.
flawed(p8): A tournament is flawed if the pairing leaves two or more playersunpaired in any round.
grade-rank(p18): An ordering by player grade.
graph(p8): A graph is formed from a set of vertices and a set of edges joiningsome of the vertices.
group(p7): For a given tournament, the set of all players of the same grade.
group g(p7): Each player in the group has grade g.
group-score(p11): The average score of all players in the group.
initial-mcmahon-score(p22): The player’s grade if player is below the bar,otherwise the bar grade.
matching(p8): A matching is a set of edges with no vertex in common.
43
maximum cardinality(p8): The matching has the largest possible number ofedges.
maximum weighted matching(p15): Each edge is given a weight, and thealgorithm chooses a matching with the largest sum of weights.
mcmahon-score(p5): The initial-mcmahon-score plus number of wins.
mcmahon-pairing(p7): Players on the same mcmahon-score are paired.
mcmahon-rank(p18): An ordering by player mcmahon-score.
mid-score(Random or Swiss)(p12): Half the number of rounds.
mid-score(McMahon)(p22): Half the number of rounds incremented by thebar grade.
opponent-grade-difference(p10): The difference between the average gradeof all opponents of a group and the group grade.
peer-games(p19): Games between players in a top-group.
perfect(p8): A pairing is perfect if every player is paired.
random(p7): Drawn from a discrete uniform distribution.
random-pairing(p6): Players are paired at random, subject to the conditionthat there are no repeat games.
split-and-cycle(p9): One or more players remain in fixed seats, the otherscycle at each round.
swiss-pairing(p7): Players with the same number of wins are paired.
tail distribution(p25): Probability X > x.
top-group(p19): Players in the range of groups from a given grade to themaximum grade.
unique winner(p5): Only one player achieves the maximum score.
vertices(p7): Edges in a graph meet in its vertices.
zero-shodan units(p7): Kyu grades are negative, dan grades start at zero,and grades increase by one unit.
44
D NOTATION
X Mean of a random variable for all samples in a trial. p11b An arbitrary bar. p19bsol The raw bar solution. p29Bsol The non-empty group nearest bsol. p29g grade. p10gmax Highest grade in the entry. p10gmin Lowest grade in the entry. p10Gc Equilibrium grade. p11G(g) Average opponent-grade for players of grade g. p10Gg(b) Average opponent grade for group g at bar b. p20∆Gg(b) Mean difference Gg(b) from g. p20∆G Mean difference of opponent-grade from group grade. p11HB Hinge point for ramp ΓB. p42HL Hinge point for ramp ΓL. p42KB Slope of ramp ΓB . p42KL Slope of ramp ΓL. p42Ψg(b) Peer-games ratio for top-group g and bar b. p19Psep Probability∆T (b) > 0 at bar solution Bsol. p35r Number of rounds. p8s(g) Group score for group g in random- or swiss- pairing. p11S(g) Mean group score for group g. p11s(g, b) Group score for group g at bar b. p22Sg(b) Mean group score for group g at bar b. p22Smid Mid score. p12Smid(b) Average scores for players at bottom of bar. p23∆S Difference mean group-score from mid-score. p12T (g, b, s) Tail distribution for given bar and grade. p25Tmid(g, b) Tail distribution for score s = Smid(b). p26ΓB(b) Linear model for Tmid(b, b). p31ΓL(b) Linear model for Tmid(b, b− 1). p42∆T (b) Bar separation. p28Θ Relative ramp distance. p35ZG Crossing point for ∆G. p13ZS Crossing point for ∆S. p13
E ALGORITHM INDEX
1. simulate-flawed-tournaments p82. simulate-result p113. swiss-pairing p164. find-bar p28
45